Lectures on Minimal Surfaces in R3

The theory of minimal submanifolds is a fascinating field in differential geometry. The simplest, one-dimensional minimal submanifold, the geodesic, has been studied quite exhaustively, yet there are still a lot of interesting open problems.

Introduction -- Definition of minimal surfaces -- The first variation -- The minimal surface equation -- Isothermal coordinates for minimal surfaces -- The Enneper-Weierstrass representation -- The geometry of the Enneper-Weierstrass representation -- Some applications of the Enneper-Weierstrass representation -- Conformal types of Riemann surfaces -- Complete minimal surfaces, Osserman's theorem -- Ends of complete minimal surfaces -- Complete minimal surfaces of finite total curvature -- Total curvature of branched complete minimal surfaces -- Examples of complete minimal surfaces -- The Halfspace Theorem and the maximum principle at infinity -- The convex hull of a minimal surface -- Flux -- Uniqueness of the catenoid -- The Gauss map of complete minimal surfaces -- The second variation and stability -- The Cone Lemma -- Standard barriers and The annular end theorem -- Annular ends lying above catenoid ends -- Complete minimal surfaces of finite topology -- Minimal annuli -- Isoperimetric inequalities for minimal surfaces -- Minimal annuli in a slab -- The existence of minimal annuli in a slab -- Shiffman's theorems -- A generalisation of Shiffman's second theorem -- Nitsche's conjecture -- Appendix: the Eigenvalue problem -- References.